# Combining Gaussian processes and polynomial chaos expansions for stochastic nonlinear model predictive control

arXiv: https://arxiv.org/abs/2103.05441 · HTML: https://mlsanity.com/paper/2103.05441/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 73 references that are in the map.
- First posted on arXiv 9 Mar 2021.

## Builds on (most cited first, 10 of 10)

- [Cautious Model Predictive Control using Gaussian Process Regression](https://mlsanity.com/paper/1705.10702/) (2017)
- [The Scenario Approach for Stochastic Model Predictive Control with Bounds on Closed-Loop Constraint Violations](https://mlsanity.com/paper/1307.5640/) (2013)
- [Polynomial-Chaos-based Kriging](https://mlsanity.com/paper/1502.03939/) (2015)
- [Stochastic data-driven model predictive control using Gaussian processes](https://mlsanity.com/paper/1908.01786/) (2019)
- [A computationally efficient robust model predictive control framework for uncertain nonlinear systems -- extended version](https://mlsanity.com/paper/1910.12081/) (2019)
- [Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints](https://mlsanity.com/paper/1410.4535/) (2014)
- [Gaussian Process Quadrature Moment Transform](https://mlsanity.com/paper/1701.01356/) (2017)
- [On the relation between Gaussian process quadratures and sigma-point methods](https://mlsanity.com/paper/1504.05994/) (2015)
- [Lyapunov-based Stochastic Nonlinear Model Predictive Control: Shaping the State Probability Density Functions](https://mlsanity.com/paper/1505.02871/) (2015)
- [Particle Model Predictive Control: Tractable Stochastic Nonlinear Output-Feedback MPC](https://mlsanity.com/paper/1612.00505/) (2016)

## Built on by (most cited first, 1 of 1)

- [Adaptive Stochastic Nonlinear Model Predictive Control with Look-ahead Deep Reinforcement Learning for Autonomous Vehicle Motion Control](https://mlsanity.com/paper/2311.04303/) (2023)

## Abstract (arXiv)

Model predictive control is an advanced control approach for multivariable systems with constraints, which is reliant on an accurate dynamic model. Most real dynamic models are however affected by uncertainties, which can lead to closed-loop performance deterioration and constraint violations. In this paper we introduce a new algorithm to explicitly consider time-invariant stochastic uncertainties in optimal control problems. The difficulty of propagating stochastic variables through nonlinear functions is dealt with by combining Gaussian processes with polynomial chaos expansions. The main novelty in this paper is to use this combination in an efficient fashion to obtain mean and variance estimates of nonlinear transformations. Using this algorithm, it is shown how to formulate both chance-constraints and a probabilistic objective for the optimal control problem. On a batch reactor case study we firstly verify the ability of the new approach to accurately approximate the probability distributions required. Secondly, a tractable stochastic nonlinear model predictive control approach is formulated with an economic objective to demonstrate the closed-loop performance of the method via Monte Carlo simulations.